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a faster way to calculate plucker coordinates that leads to quadratic algebra

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A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

A faster way to calculate Pl¨cker Coordinates u

March 7, 2011

A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

The Pl¨cker Coordinate is a unique vector associated with u every k-dimensional space in an n-dimensional space. Thus, they can be described as an embedding in a higher, namely the (n )-dimensional space. To briefly describe Pl¨cker coordinates, u k they are calculated as follows. Form a matrix with the basis of the k-space whose coordinate we are to find. Calculate the (n ) k k × k minors that result from choosing any k columns, and stack them all up in some defined order. the resulting (n ) k dimensional vector is called the Pl¨cker coordinate of that u subspace.

A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

Now, in order to calculate the Pl¨cker coordinate of a general u k-space in an n-space, we would have to calculate (n ) k × k k minors. The method proposed relies on the ordering of the basis of the given k-space, and the fact that we are storing the elements that have been calculated.

A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

Here is a test case to illustrate the same. Let us use look at 3-dimensional subspaces in a 6-dimensional space. Assuming the basis set can be represented as follows:   1 0 0 a1 b1 c1 0 1 0 a2 b2 c2  0 0 1 a3 b3 c3

A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

Now, let us order the minors to be calculated as follows:
1

An example Comparison Calculation of order time

First, choose the first 3 columns, i.e. the identity matrix. This gives us the minor 1. That is the first minor. Next, replace only one column, namely the column 3, by column a, and continue on till column 1 gets replaced by column c. That makes it a total of 9 more minors. Next, replace 2 columns, by which we will get 9 more minors. Finally, all the columns are replaced. that makes it 1 + 9 + 9 + 1 = (n ) = 6 = 20 minors. 3 k

2

3

4

A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

The first minor : The first minor is just the determinant of identity, and therefore equal to 1. The second set : The next set of minors are as follows. Since we replace only one column, only one entry is of significance, the rest obliterated by the zeroes in the remaining columns, thus the entry we are looking for is the i th entry of the new column that replaced the i th column. This can be seen in the following example, Lets replace the 3rd column with column b. Then the minor we are looking at is:   1 0 b1 0 1 b2  0 0 b3 By inspection, this comes out to b3 .
A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

The third set : The next set of minors are those where 2 columns are replaced. Let the 1st and the 3rd be replaced by b and c. Then the resulting matrix is   b1 0 c1 b2 1 c2  b3 0 c3 Now, expanding along the columns that are not replaced, we get b1 c3 − c1 b3 That is precisely the determinant of the 2 × 2 matrix got by using the i th element of the column replacing the i th column, and similarly for the j th . Note that after getting all such 2 × 2, we have them stored as part of the Pl¨cker coordinate. u
A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

The fourth set : The next set is simply all the columns replaced, in our test case, i.e.   a1 b1 c1 a2 b2 c2  a3 b3 c3 Heres where the method finally gets useful. In the conventional way, we would have to calculate the determinant as a1 (b2 c3 − c2 b3 ) − a2 (b1 c3 − c1 b3 ) + a3 (b1 c2 − c1 b2 )

A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

Now, if we look at the terms in parenthesis, they are just 2 × 2 determinants that are already calculated. Thus the proposed method calculates the determinant as a1 (pp ) − a2 (pq ) + a3 (pr ) where p, q, r are elements of the Pl¨cker coordinate that we u have already calculated. As we can see, the bigger the matrix, the more time the method saves!

A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

In general, for an m × m minor, while the conventional way would take the order of time to calculate an m × m determinant, which is O(m!) if calculated naively or O(m3 ) using methods like LU decomposition, while the proposed method would do it as x1 (px1 ) − x2 (px2 ) + . . . + xm (pxm ) where the pi are elements of the Pl¨cker coordinate that are u already calculated. Thus, the proposed method takes O(m) time for the same minor.

A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

An example Comparison Calculation of order time

Let us try to calculate the time required by both methods, given that the we are working with n dimensional subspaces. Separating by number of columns replaced, 0→1 1 → (n ) 1 2 → (n ) 2 ... n → (n ) n Let us replace O(m) as m for ease of calculation. Thus, the steps we are looking at can be approximated as (n ) i 3 i vs (n ) i i
A faster way to calculate Pl¨cker Coordinates u

A faster way to calculate Pl¨cker u Coordinates

Using (1 + x)n = Differentiate to get (n ) x i i

An example Comparison Calculation of order time

n ∗ (1 + x)n−1 = Substitute x = 1 to get

(n ) i ∗ x i−1 i

(n ) i = n ∗ 2n−1 i Similarly, differentiate twice more, and approximately we should get (n ) i 3 = n3 ∗ 2n−3 i Thus, we see that the proposed method indeed speeds things up.
A faster way to calculate Pl¨cker Coordinates u

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